doi:10.14311/AP.2017.57.0272
Ac a Poly echnica 57(4):272–281, 2017 ©Czech Technical Uni e si y in P ague, 2017
a ailable online a h p://ojs.c u .cz/ojs/index.php/ap
AERODYNAMIC LOAD OF AN AIRCRAFT WITH A HIGHLY
ELASTIC WING
Pa el Schoř
Ins i u e o ae ospace Enginee ing, Facul y o Mechanical Enginee ing ,B no Uni e si y o Technology ,
Technicka 2896/2, 616 69 B no Czech Republic
co espondence: [email p o ec ed]
Abs ac . In his a icle, a me hod o calcula ion o ai loads o an ai c a wi h an elas ic wing is
p esen ed. The me hod can p edic a edis ibu ion o ai loads when he elas ic wing de o ms. Unlike
he adi ional Eule o Na ie -S okes CFD o FEM coupling, he me hod uses 3D panel me hod as
a sou ce o ae odynamic da a. This makes he calcula ion easible on a ypical ecen wo ks a ion.
Due o a sho compu a ional ime and low ha dwa e demands his me hod is sui able o bo h he
p elimina y design s age and he load e alua ion s age. A case s udy is p esen ed. The s udy compa es
a glide wing pe o ming a pull maneu e a bo h igid and and elas ic s a e. The s udy indica es a
signi ican edis ibu ion o ai load a he elas ic case.
Keywo ds: ae odynamics, panel me hod, ini e elemen me hod, wing load, luid s uc u e in e ac ion.
1. In oduc ion
1.1. P oblem o e iew
A ligh o a mode n glide o a High Al i ude Long
Endu ance (HALE) ai c a is a ep esen a i e exam-
ple o he luid-s uc u e in e ac ion p oblem. These
lying ehicles a e a esul o a sea ch o he mos
ae odynamically e ec i e shape. This leads o a need
o a wing wi h a la ge wingspan and an aspec a io.
Mos ecen glide s ha e a wingspan anging om 18
o 30 me e s wi h an aspec a io app oxima ely om
30 o 50. A NASA Helios HALE ai c a had a wing
wi h a wingspan o 75 m and aspec a io o 30 [
1
].
Du ing a ligh o such ai c a , he wing unde goes
la ge de o ma ions, s ill in he elas ic egime, as shown
on Figu e 1.
The Helios HP-03 expe imen al HALE ai c a
c ashed on June 26, 2003 a e encoun e ing a mo-
sphe ic u bulence. The ai c a de eloped a wing
Figu e 1. The Helios ai c a unde going la ge de o -
ma ion [1].
ip displacemen o mo e han 12
m
and consecu i e
pi ch oscila ions, esul ing in s uc u al ailu e o he
leading edge s uc u e, bu lea ing he main spa un-
damaged. A e he c ash o Helios HP03 p o o ype,
see Figu e 2, a demand ose o new me hods, which
a e capable o compu e loads o highly elas ic, “mo -
phing” ai c a s [
1
]. In his a icle, a me hod o a
solu ion o quasi-s a ic load cases o such ai c a is
p esen ed.
Mo eo e , ecen ai wo hiness egula ions explici ly
equi e o calcula e he load edis ibu ion in case “i
de lec ions unde load would signi ican ly change he
dis ibu ion o ex e nal o in e nal loads” [2, 3].
1.2. Me hods o e iew
Va ious me hods a e adi ionally used in o de o
cap u e he change o he ae odynamic load gene a ed
in igid and elas ic s a e. The simples op ion is o
combine a o ex la ice me hod [
4
] wi h a simple
beam ini e elemen model [
5
] . The mos sophis i-
ca ed app oach consis s o Na ie -S okes CFD sol e
Figu e 2. The Helios ai c a a e s uc u al ailu e
du ing ligh [1]
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ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing
o ae odynamic inpu combined wi h a de ailed FEM
s uc u al model wi h solid elemen s [
6
]. The ae ody-
namic load is hen di ec ly in e pola ed o he FEM
model. Bo h o hese app oaches need o i e a e be-
ween he ae odynamic and s uc u al model o ob ain
he equilib ium. The simple model wi h he li ing
line heo y p o ides he as es solu ion imes, how-
e e , e o s may be in oduced because o an imp ope
pi ching momen dis ibu ion due o he simpli ied
VLM ae odynamic model. The de ailed model wi h
he Eule o Na ie -S okes CFD sol e wi h a de ailed
FEM s uc u al model can p o ide he mos accu a e
p edic ion o he load edis ibu ion. Typically, he
CFD olume mesh has o be upda ed acco ding o
he esul s om he FEM solu ion. This may cause
issues wi h he de o ma ion o he CFD olume mesh
when he de o ma ion o he wing is la ge. Recen ly
an Imme sed Bounda y Me hod is used o sol e his
kind o luid-s uc u e in e ac ion [7]
When assuming he cos s o he compu a ional ime
o he CFD sol e and also assuming ha up o 10
i e a ions a e usually equi ed o ob ain he con e ged
solu ion o one pa icula angle o a ack, he use
o he CFD sol e is es ic ed o esend high pe -
o mance compu e clus e s, he e o e making i e y
expensi e and no easible o a ypical glide manu ac-
u e nei he a a design s age no a a load e alua ion
s age.
In his a icle, a 3D panel me hod is used, combined
wi h a geome ically nonlinea ini e elemen me hod
wi h beam elemen s o model he s uc u e o he wing.
The sa ings o compu a ional ime when compa ed o
he CFD N-S sol e and he di e ences in compu ed
ai loads jus i y he use o his low ideli y ae odynamic
ool, making i possible o compu e la ge numbe o
load cases on an a e age wo ks a ion.
1.3. Eligibili y
Based on assump ions made in Sec ions 2.1 and 2.2,
he me hod p esen ed in his a icle is sui able o
analysis o quasi-s a ic maneu e s o an ai c a wi h
elas ic wings, ope a ing a a low Mach numbe (
M <
0
.
3) and su icien ly high Reynolds numbe (
Re >
1000000). This includes he HALE ai c a du ing
i s low al i ude ligh o a glide pe o ming a pull
maneu e .
2. Me hods
2.1. Ae odynamic model
A luid mo ion is adi ionally desc ibed by se o equa-
ions called Na ie -S okes equa ions [
8
]. The numbe
o he exac solu ion o Na ie -S okes equa ions is
small. Fo p ac ical p oblems, hese equa ions a e
sol ed nume ically [9].
Sol ing he Na ie -S okes equa ions p o ides a de-
ailed desc ip ion o he low, howe e he compu a-
ional cos s a e eno mous. The e o e some assump-
ions a e made o educe he complexi y o he p ob-
lem. I he low is assumed o be s eady, in iscid,
Figu e 3. Po en ial low domain.
incomp essible and i o a ional, is called a po en i al
low and i can be desc ibed by Laplace’s equa ion [
8
]
:
∇2Φ=0 (1)
Whe e Φis a scala unc ion called eloci y po en ial
and eloci y
−→
q
a each poin can be ob ained by i s
de i a ion:
−→
q=∇Φ(2)
An app oach om [
10
] is used o sol e he Laplace’s
equa ion. Assume a c oss sec ion o a wing as shown
in Figu e 3. The su ace
S∞
encloses he p oblem a
in ini y, su ace
S
ep esen s a body (wing) and su ace
W
ep esen s i s wake. The su ace
S
+
W
di ides he
domain in o wo egions: he ex e nal egion wi h he
low ield o in e es and eloci y po en ial Φand he
in e nal egion wi h he ic i ious low and eloci y
po en ial Φ
i
. The su aces a e modeled by a double
and sou ce singula i ies.
G een’s heo em is applied o bo h he ou e and
inne egion and eloci y po en ial Φ
P
is ob ained by
combining bo h exp essions. [
10
] The eloci y po en-
ial Φ
P
gi es a eloci y po en ial anywhe e in he wo
egions. I is exp essed in e ms o su ace in eg als in
e ms o eloci y po en ial and i s no mal de i a i e
o e he bounda y su ace as
ΦP=1
4πZZS+W+S∞
(Φ −Φi)−→
n· ∇ 1
dS
−1
4πZZS+W+S∞
1
−→
n·(∇Φ− ∇Φi) dS. (3)
Whe e
is he dis ance om he poin
P
o he
elemen d
S
on he su ace, and
−→
n
is a uni no mal
ec o o he elemen d
S
. [
10
] The i s in eg al in (3)
ep esen s he dis u bance po en ial om a su ace
273
Pa el Schoř Ac a Poly echnica
dis ibu ion o double s wi h a densi y o (Φ
−
Φ
∞
).
The second in eg al ep esen s he con ibu ion o
sou ces wi h a densi y o −−→
n·(∇Φ− ∇Φi).[10]
Nex , he in e nal Di ichle bounda y condi ion
is in oduced. This bounda y condi ions se s he
In e nal low equal o he Onse low by
Φi= Φ∞,(4)
ΦP= Φ∞.(5)
Following he p ocedu e om [
8
,
10
,
11
], he su ace
is disc e ized in o
n
su ace panels and
nw
panels in
he wake egion wi h a cons an dis ibu ion o singu-
la i ies. The (3) is also disc e ized and he Di ichle
bounda y condi ion is e alua ed a cen oid o each
su ace panel. These poin s a e called Colloca ion
poin s. The esul o his p ocedu e is a linea equa-
ion [8]
n
X
k=1
Ckµk+
nw
X
l=1
Clµl+
n
X
k=1
Bkσk= 0,(6)
whe e
Ck
,
Cl
a e double in luence coe icien s o body
and wake panels,
Bk
is he sou ce in luence coe icien ,
µk
is he double s eng h o a body panel,
µl
is he
double s eng h o a wake panel,
σk
is he sou ce
s eng h o a body panel. The σkis se as
σk=−→
nk·−−→
Q∞,(7)
whe e
−→
nk
is he panel no mal uni ec o and
−−→
Q∞
is
he ees eam eloci y.
The me hod desc ibed abo e is known as he “ i s
o de panel me hod” [
8
]. I is used as he only sou ce
o ae odynamic da a o he luid-s uc u e in e ac-
ion. The o mula ion o he panel me hod is iden-
ical o well-known and alida ed codes such as he
VSAERO [10] and PMARC [11].
Bounda y laye .
The po en ial low may gi e use-
ul esul s o some applica ions, howe e o applica-
ions whe e iscous e ec become mo e impo an , he
esul s ob ained unde assump ion o po en ial low
may become inaccu a e in e ms o o e p edic ed li
cu e slope. A ypical example o a low wi h a signi -
ican iscous e ec is a glide wing a a low Reynolds
numbe o a low o e a mul i-elemen wing.
The iscous e ec s can be in oduced in o he po en-
ial low model by displacing he ac ual body su ace
by a displacemen hickness o he bounda y laye [
12
].
The ac ual displacing o he body su ace is pe -
o med by a modi ica ion o panels sou ce s eng h
in (7) o
σk=−→
nk·−−→
Q∞+d
ds(ueδ∗),(8)
whe e sis a dis ance on he body, ueis a panel edge
eloci y and
δ∗
is a displacemen hickness compu ed
by he bounda y laye analysis [12].
2.2. S uc u al model
The s uc u e o he wing is modeled using he FEM
wi h beam elemen s only. Each node o he elemen
has six deg ees o eedom. The s i ness p ope ies
a e assumed cons an o each elemen . The s i ness
ma ix o such elemen can be ound in many FEM
ela ed ex books [
13
]. Howe e such elemen s a e
usually de i ed o iso opic ma e ials. In ac , wings
o mode n glide s a e made o a ca bon ibe ein-
o ced composi e o ho opic ma e ial. A wing made
o such ma e ial exhibi s coupling be ween bending
and o sion. This causes issues, when one ies o
educe s i ness p ope ies o a composi e wing in o
single s i ness ma ix o a beam elemen . A me hod
om [14] was adop ed o sol e his issue.
As he wing unde goes la ge de o ma ions i is nec-
essa y o cap u e he geome ical non linea i y. To
sol e his issue a New on/Rhapson i e a i e me hod is
used. The geome ical non-linea i y, howe e , causes
di icul ies when calcula ing elemen s s i ness ma i-
ces — one mus de e mine he de i a ion o he global
s i ness ma ix ( angen s i ness ma ix). A me hod
om [
15
] was adop ed, which p o ides a nume ically
gene a ed angen s i ness ma ix o uss and beam
elemen s.
2.3. Ae o-s uc u al in e ac ion
Wo k low.
A b ie o e iew o he in e ac ion be-
ween he ae odynamic and in e nal o ces in he wing
s uc u e is shown on Figu e 4. The in e ac ion is
done by using a modi ied New on-Rhapson me hod.
This me hod and s eps called he “Compu e ae ody-
namic o ces” and “Compu e wing displacemen ” om
Figu e 4 a e explained in de ail in he pa ag aph Load
con ol scheme. Compu ing he igid li cu e as a
i s s ep is no necessa y, he main eason o his
is o ind he ze o li angle o he a ack
α0
, whe e
he compu a ion o elas ic li cu e s a s. S a ing
om
α0
makes he wing almos unloaded and he
New on-Rhapson me hod is likely o con e ge. Also,
he inc emen o he angle o a ack ∆
α
should be
small enough o assu e he con e gence, i is usually
om 0.5° o 2°.
In e ac ing elemen .
The in e ac ing elemen is
explained in Figu e 5. I consis s o ae odynamic
panels, bu he e is only one elemen in he span-wise
di ec ion and one FEM beam elemen . Fo ces and
momen s a e always in eg a ed om he wing ip o
plane o symme y, a a poin which is also a node
o he ini e elemen model. The load applied a he
FEM node is a sum o o ces momen s om all panels
belonging o he in e ac ing elemen as shown on
Figu e 5:
−→
=
n
X
i=1
−→
i,(9)
−→
m=
n
X
i=1
−→
i× i,(10)
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ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing
Figu e 4. Ae o-s uc u al in e ac ion — wo k low.
whe e
i
is he o ce on
i
h panel in he global coo -
dina e sys em
x, y, z
, as shown in Figu e 5;
i
is he
ec o be ween he e e ence node and con ol poin
o i h panel.
The esul o he FEM analysis is a displacemen
ec o u a each node. Coo dina es o ae odynamic
panels a e upda ed a each load s ep using nodal
ansla ional and o a ional displacemen s, acco ding
o (11). Also, he nodal coo dina es a e upda ed as
explained in [15].
PTSnew =Rx(∆φx)Ry(∆φy)Rz(∆φz)PTSold
+ ∆U, (11)
whe e ∆
φx
,∆
φy
,∆
φz,
a e o a ional inc emen s in
he global coo dina e sys em calcula ed om he las
s ep o he New on-Rhapson i e a ion, see bellow; ∆
U
is he displacemen inc emen ;
Rx
(
φx
)
, Ry
(
φy
)
, Rz
(
φz
)
a e o a ional ma ices de ined by
Rx(∆φx) = 

1 0 0
0 cos φx−sin φx
0 sin φxcos φx

,(12)
Ry(∆φy) = 

cos φy0 sin φy
0 1 0
−sin φy0 cos φy

,(13)
Rz(∆φz) = 

cos φz−sin φz0
sin φzcos φz0
0 0 1

.(14)
Figu e 5. Panel me hod — FEM in e ac ion.
Load con ol scheme.
In he New on/Rhapson
me hod, he o al load is di ided in o m load s eps
and o each load s ep, a New on i e a ion is done
un il an equilib ium be ween he in e nal o ces and
ex e nal load is ound. A e ha , he ex e nal load
is inc eased by he load s ep. Howe e , in he case
o an elas ic wing, he o al ex e nal load is unknown
because he load (li ) also depends on how much
he wing is de o med. The e o e, he s anda d load
con ol in he New on/Rhapson me hod was changed
in he ollowing way: The load is con olled by in-
c easing he angle o a ack as explained in Figu e 4.
Usually he li gene a ed by he wing (sailplane) is
he dependen a iable in he
u
–
F
cu e. To compu e
he load gene a ed by he elas ic wing, a ollowing
scheme is used:
(1.)
The i s i e a ion s a s a an angle o a ack o
ze o li o a igid wing. The equilib ium be ween
he ex e nal load compu ed om he panel me hod
and in e nal o ces is compu ed using he New on
i e a ion.
(2.)
The angle o a ack is inc eased by a small s ep,
e.g., 1
°
and he equilib ium be ween he ae ody-
namic load and he in e nal o ces is compu ed by
he New on i e a ion.
(3.)
The angle o a ack is u he inc eased un il he
o al load (li ) is equal o he desi ed load (li ).
The esul is a nonlinea li cu e. Fo each load
s ep, he ec o o in e nal o ces is also a nonlinea
unc ion o p e ious load s eps. The e o e when com-
pu ing a load o an elas ic wing i is con enien o
compu e a wing li cu e as desc ibed abo e wi h
e y small inc emen o he angle o a ack, e.g., 0
.
5
°
and o sa e he ec o o de o ma ion and in e nal
o ces o each con e ged load s ep. When compu -
ing la ge numbe o load cases, signi ican sa ings
in he compu a ional ime a e achie ed when he
p e-compu ed de o ma ion ec o and in e nal o ces
ec o a e loaded o he closes lowe angle o a -
ack. One hen needs o compu e only he di e ence
be ween he p e-compu ed angle o a ack and he
desi ed angle o a ack.
275
Pa el Schoř Ac a Poly echnica
Figu e 6. Ae o-s uc u al in e ac ion — load con ol in New on-Rhapson me hod.
Figu e 7. Coo dina e sys ems.
3. Nume ical s udy
3.1. O e iew
A nume ical s udy was conduc ed o demons a e he
impo ance o he elas ici y o he wing o i s load.
Assuming a ligh en elope om [
2
, pa . CS22.333],
and conside ing he poin
A
o he en elope o be c i -
ical, he ae odynamic load a his poin is e alua ed
using h ee di e en me hods:
(1.) panel me hod wi h elas ic wing;
(2.) panel me hod wi h igid wing;
(3.) li ing line heo y.
The li ing line heo y is used as a alida ion ool o
he panel me hod.
3.2. Coo dina e sys ems
A global igh -handed coo dina e sys em
OXY Z
de-
ined in Figu e 7 is used. Addi ional local coo dina e
sys ems
oixiyizi
a e used o e alua ion o he local
load. As he wing de o ms, hese coo dina e sys ems
P ope y Symbol Value
A ea A 11.55 m2
Span L 21.0 m
Roo cho d c 0.7 m
Tip cho d c 0.4 m
Dihed al Γ0 ad
Sweep Θ0 ad
Table 1. Wing geome y.
P ope y Symbol Uni
C oss sec ion a ea ASm2
Second momen o a ea Ixm4
Second momen o a ea Izm4
To sional cons an Jym4
O se o neu al axis ∆Xnm
O se o neu al axis ∆Znm
O se o elas ic axis ∆X m
O se o elas ic axis ∆Z m
Young modulus EN/m2
Shea modulus GN/m2
Table 2. Wing c oss sec ional p ope ies.
ollow he de o med geome y. The ee s eam eloc-
i y
V
wi h posi i e angle o a ack is also shown in
Figu e 7.
3.3. Wing specimen
A simple ape ed wing is examined in his s udy. The
wing is designed as a plana wing in unde o med s a e.
The wing has one s aigh main spa loca ed a 0
.
25
c
,
whe e
c
is local cho d. The main eason o his
geome y is he e alua ion o he load by he li ing
line heo y, which can no be used o a non-plana
wing geome y and no swep wings.
276

ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing
Figu e 8. Spanwise a ia ion o Iz,Ix,Jy.
Figu e 9. Ai oil E603 in iscid ae odynamic da a.
Geome y.
Geome y o he wing is desc ibed in
Table 1. An Epple E603 ai oil is used o all sec ions
o he wing.
S i ness p ope ies.
When analyzing he de o -
ma ion o he wing by he FEM analysis desc ibed
abo e, an ins umen al s ep is he de e mining o all
c oss-sec ional cha ac e is ics de ined in Table 2 o
all c oss sec ions o he wing. In his example, he
Young’s modulus
E
and shea modulus
G
a e assumed
as cons an o he whole wing wi h ollowing alues:
E= 60 GPa, G = 7 GPa.
The wing has only one spa a 0
.
25
c
and he nodes a e
coinciden wi h he spa . The e o e he o se o he
elas ic axis is almos negligible and bo h alues a e
assumed o be ze o. The neu al axis is loca ed a he
local cen oid wi h non-ze o o se s, bu hese o se s
a e no shown in his pape . The spanwise a ia ion
o he Second momen s o a ea and o sional cons an
is shown on Figu e 8.
Fini e elemen model.
The wing is di ided in o
162 spanwise sec ions. Each sec ion has one wo-
noded beam elemen desc ibed in Sec ion 2. Nodal
coo dina es always co espond o a loca ion o a main
spa . The node loca ed in he plane o symme y has
all deg ees o eedom p esc ibed o ze o. All coupling
e ms in he elemen s i ness ma ices ela ed o he
bending- o sion coupling a e in en ionally se o ze o
o cla i y.
Ai oil da a.
The ae odynamic cha ac e is ics o
he Epple E603 ai oil we e ob ained om he XFOIL
in iscid analysis, as shown in Figu e 9. This ae ody-
namic da a a e used only o he li ing line analysis.
3.4. Loadcase VA
Assuming a maximum li coe icien o he wing
CLmax
= 1
.
4, om Figu e 10, he ligh condi ions a
poin
A
o he maneu e ing en elope a e shown in Ta-
ble 3 The o al mass o he glide is
MTOW
= 750 kg,
bu ine ia o ces ac ing on he wing a e no assumed
in he analysis o cla i y.
277
Pa el Schoř Ac a Poly echnica
Figu e 10. Compa ison o igid and elas ic wing — li .
P ope y Symbol Value
Ai speed V64.0 m/s
Wing li coe icien CL 1.4
Load ac o n5.3
Table 3. Load case VA.
De o ma ion sequence.
De o ma ion sequence o
a ious wing li coe icien is shown in Figu e 11. I
mus be no ed ha his de o ma ion sequence is alid
only o ai speed
V
= 64
.
0m/s. As he de o ma ion
o he wing depends on o ces and momen s applied
on he wing su ace, o a di e en ai speed, he co -
esponding de o med shape will be di e en .
In eg al ae odynamic cha ac e is ics.
In eg al
ae odynamic cha ac e is ics o he igid and elas ic
wing a e compa ed on Figu es 10 and 12. A close
ag eemen o igid cases can be obse ed.
Fo he elas ic case a signi ican dec ease o he
slope o he wing li cu e can be seen in Figu e 10.
Addi ionally, he inc ease o pi ching momen o he
elas ic wing can be seen in Figu e 12. The di e -
ences be ween igid and elas ic cases a e discussed in
Sec ion 4
Load compa ison.
Ac ual compa ison o he load
o he igid and elas ic wing pe o ming a s eady ma-
neu e wi h he same load ac o
n
= 5
.
3is shown in
Figu es 13–15. Only bending and o sional momen s
a e shown. The di e ences be ween he igid and
elas ic case a e discussed in Sec ion 4
4. Discussion
4.1. Valida ion o esul s
The e a e wo main op ions o alida e he esul
compu ed by he p esen ed me hod:
Figu e 11. De o ma ion sequence a VA.
Nume ical alida ion.
A nume ical alida ion
means o p e o m he case s udy using a RANS CFD
sol e as a sou ce o ae odynamic da a coupled o a
wing modeled by shell elemen s o by 3D solid ele-
men s.
Expe imen al alida ion.
Expe imen al alida-
ion means o ac ually ly a glide , pe o m he pull
maneu e and measu e he displacemen o he wing.
Conclusion on alida ion.
Un o una ely bo h
nume ical and expe imen al alida ion would equi e
much mo e e o and unds han wha was spen o
his whole wo k. A li e a u e e iew also did no gi e
any example, which could be used as a alida ion
benchma k. The e o e, his wo k is p esen ed only as
a nume ical s udy wi hou any o he alida ion.
4.2. Dec ease o wing li cu e slope
As he li gene a ed by he elas ic wing inc eases, he
wing ben s mo e, as shown in Figu e 11. As he wing
ben s mo e, he slope o he li ing cu e d ops. This
is demons a ed in Figu e 10, which shows a educed
slope o he li cu e o he elas ic wing, compa ed
o he igid wing. This phenomena could possibly be
explained by he educ ion o he p ojec ed a ea o
278
ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing
Figu e 12. Compa ison o igid and elas ic wing — pi ching momen .
Figu e 13. Compa ison o igid and elas ic wing — No mal bending momen Mx.
he wing and he change o di ec ion o he li o ce.
This has ollowing consequences:
(1.)
When he p esc ibed o al li is eached, he
wing ope a es a a highe angle o a ack. In he
pa icula case p esen ed in Sec ion 3 he di e ence
o he c i ical angle o a ack is 6°.
(2.)
The no mal bending momen is sligh ly educed,
see Figu e 13. The angen ial bending momen is,
howe e , inc eased due o he inc eased angle o a -
ack, see Figu e 15. In he pa icula case p esen ed
in Sec ion 3 he angen ial bending momen o he
elas ic wing is 1
.
34 imes highe , han in he igid
case. The no mal bending momen is, howe e , 0
.
87
imes lowe .
4.3. Inc ease o wing pi ching momen
The explana ion o he inc ease o he wing pi ching
momen equi es mo e labo han he explana ion o
he dec ease o he wing li cu e slope. Fi s , assume
a wing c oss sec ion loca ed a he wing ip. The
wing is o an unde o med shape and ope a es nea
he c i ical angle o a ack. The sec ion gene a es
a pi ching momen a ound he main spa
MY 1
=
1
2ρ 2S1c1cm1
(
α
). The esul an o he ae odynamic
o ce is assumed o ac a he main spa (0
.
25
c
). The
di ec ion o he o ce is upwa d and o wa d (nega i e
X
, posi i e
Z
). When he wing has ze o dihed al
and ze o sweep, he con ibu ion o his o ce o he
pi ching momen is ze o, since he ec o o he sec ion
e e ence poin ela ed o he wing e e ence poin
~ 1
= [0
,−
10
.
37
,
0]. The wing e e ence poin is de ined
as a poin (node) on he main spa in he XZ plane.
Nex , assume a highly de o med wing shown in
Figu e 16. The wings ip sec ion is displaced by a
ec o
~
d1= [−0.12,2.5,5.15],
he ec o o he sec ion e e ence poin ela ed o he
wing e e ence poin is now
~ 1= [−0.3,−8,5.15].
The esul an o he ae odynamic o ce a he wing
279
Pa el Schoř Ac a Poly echnica
Figu e 14. Compa ison o igid and elas ic wing — To sional momen My.
Figu e 15. Compa ison o igid and elas ic wing — Tangen ial bending momen Mz.
e e ence poin is compu ed as
~
1= [−6.07,49.38,9.31],
hen, he esul ing momen a he wing e e ence poin
gene a ed by he o ce ~
1is
~m1=~ 1×~
1= [−329.05,−28.51,−63.32].
When he pi ching momen o he sec ion is e alu-
a ed, he con ibu ion o ∆
MY 1
=
−
28
.
51 Nm causes
o inc ease he pi ching momen coe icien o he sec-
ion by a alue o
−
0
.
55 mos ly due o he e ical
displacemen o he sec ion and he o ce esul an
di ec ing o wa d.
4.4. Wing maximum li
In his a icle, he maximum li coe icien o he igid
wing was compu ed using he nonlinea li ing line
heo y as
CLmax
= 1
.
4. The ai oil sec ional da a used
in he analysis we e compu ed by he XFOIL in iscid
analysis, assuming o be close o he expe imen al
da a. Since he pu pose o his a icle is o p esen a
Figu e 16. Pi ching momen o highly de o med wing.
po en ial p oblems o an ai c a wi h a highly elas ic
wing, a he han o make a p esc ip ion based on
p ac ical esul s, he maximum li coe icien o he
elas ic wing is also assumed o be CLmax = 1.4.
280